A Hilton-Milner-type theorem and an intersection conjecture for signed sets

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, ..., n} and any integer k ≥ 2, let Sn,r,k be the family {{(x1, y1), ..., (xr, yr)} : x1, ..., xr are distinct elements of [n], y1, ..., yr ∈ [k]} of k-signed r-sets on [n]. Let m := max{0, 2r − n}. We establish the following HiltonMilner-type theorems, the second of which is proved using the first: (i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1|+ |A2| ≤ ( n r ) k − r ∑ i=m ( r i ) (k − 1) ( n− r r − i ) kr−i + 1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2 ≤ r ≤ n, then |A| ≤ { (n−1 r−1 ) kr−1 − ∑r−1 i=m ( r i ) (k − 1)i ( n−1−r r−1−i ) kr−1−i + 1 if r < n; kr − (k − 1)r + r − 1 if r = n. We also determine the extremal structures. (ii) is a stability theorem that extends Erdős-Ko-Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.